System and method for resolving phase ambiguity of a transducer array to determine direction of arrival of received signals

ABSTRACT

System and method for resolving phase ambiguities of various transducer arrays, including non-coplanar interferometer antennas on an aircraft skin, in order to determine the direction of arrival of a signal received by the array and emitted by a source remote from the array.

FIELD OF THE INVENTION

The invention generally relates to direction finding systems andmethods. In particular, the invention includes a system and method forresolving phase ambiguities of various transducer arrays, includingnon-coplanar interferometer antennas on an aircraft skin, in order todetermine the direction of arrival or angle of arrival of a signalreceived by the array and emitted by a source remote from the array.

BACKGROUND OF THE INVENTION

The prior art contains several methods of phase ambiguity resolution forinterferometric systems consisting of either collinear or non-collinear,coplanar arrangements of transducers, such as antennas. Referring toFIG. 1, in the case of an antenna array, which is sensitive toelectromagnetic radiation, antenna elements A1 and A2 of the antennaarray are presented with an electromagnetic wave emitted by a remotesource. The wave is incident at the “phase centers” of each of theelements A1, A2 of the array from the exact same direction. Thisdirection is referred to as either the direction of arrival (DOA) or theangle of arrival (AOA). Phase ambiguities arise under conditions inwhich the two antennas are further apart than one half wavelength of thesignal carrier wave because practical phase comparators are incapable ofdiscerning a phase angle outside of the range of ±π (±180°).

In a treatise published in 1973 by James E. Hanson titled “On ResolvingAngle Ambiguities of n-Channel Interferometer Systems for ArbitraryAntenna Arrangements In a Plane” (Defense Technical Information CenterPublication Number AD 776-335) addresses ambiguities. In this treatise,Hanson demonstrated how the problem of interferometric phase ambiguityresolution could be easily approached by casting the severaldifferential phase measurements into direction cosine space as a set ofequally spaced parallel straight lines; these straight lines arise fromrecasting the interferometer equation as a linear equation:$\begin{matrix}{{\psi = {{\frac{2\pi\quad d_{y}}{\lambda}\sin\quad\phi\quad\sin\quad\theta} + {\frac{2\pi\quad d_{z}}{\lambda}\cos\quad\theta} - {2\pi\quad k}}}{{k = 0},{\pm 1},{\pm 2},\ldots\quad,}} & (1)\end{matrix}$

wherein:

Ψ is the measurable differential phase;

λ is the electromagnetic wavelength;

φ is the azimuth angle;

θ is the zenith angle;

k is an integer chosen to make Ψ come out in the range of ±π; and

d_(y) and d_(z) are the y and z components of the inter-element baselinevector.

The meanings of the terms involved in equation (1) are illustrated inFIG. 1. In Hanson's representation sin φ sin θ and cos θ are replaced byY and Z, respectively, and the new equation is manipulated so that itappears as: $\begin{matrix}{Z = {{{- ( \frac{d_{y}}{d_{z}} )}Y} + {\frac{\lambda}{2\pi\quad d_{z}}{( {\psi + {2\pi\quad k}} ).}}}} & (2)\end{matrix}$

Equation (2) is the equation of a set of parallel straight lines, oneline for each value of the integer k. In addition, Hanson defines a unitcircle as:(sin φ sin θ)²+(cos θ)²=1.  (3)

This unit circle describes the limits of visible space in thateverywhere on and inside this unit circle (sin φ sin θ)²+(cos θ)²≦1.Accordingly, it is referred to as the unit circle of visibility. Thesesets of parallel lines along with the unit circle centered in directioncosine space are known to those familiar with Hanson's work as Hansonambiguity diagram and the sets of straight lines are referred to asHanson ambiguity trajectories. The entire set of trajectories completelydescribe the ambiguity performance of a linear or a non-linear, coplanarinterferometer array (see FIGS. 2 and 2B).

According to Hanson, phase ambiguity resolution is accomplished byfinding an arrangement of three or more antennas that create a Hansonambiguity diagram with but a single point of intersection of the varioustrajectories, an intersection that is located in direction cosine spaceat the exact position of the radiating source; for strictly collineararrays of antennas the single intersection is rather a single straightline. It is also noted that this single point of intersection indirection cosine space leads immediately to the two angles of arrival—φthe azimuth angle and θ the zenith angle—so that ambiguity resolutionleads immediately to the determination of the angles of arrival (seeFIG. 2A).

The differential phase measurements made with practical interferometerscome with errors that arise due to systematic as well as thermodynamicperturbations within the array antennas and the receiving network. Theseerrors cause the Hanson trajectories to move or shift randomly at rightangles to the directions in which they lay. As a consequence, the singlepoint of intersection in the ideal, no error condition becomes a set ofpair-wise trajectory intersections (see FIG. 3). Thus, ambiguityresolution is accomplished by designing the ambiguity resolutioncomputer algorithm so that it can discern a tightly grouped set ofpair-wise intersections. Such an approach is described by Azzarelli, etal. in U.S. Pat. No. 6,140,963 but only for non-linear, coplanar arrays.

However, there is a need for a system and an ambiguity resolution methodwhich can deal with non-coplanar arrangements of antenna elements. Inaddition, there is a need for a system and method which deal with thephase errors that arise due various perturbations and which deal withother than ideal conditions.

SUMMARY OF THE INVENTION

The invention includes a system and method for resolving the angularambiguities inherent in the differential phase measurements of aninterferometric system of non-coplanar antennas. The system and methodare also applicable to other transducer systems, such as an underwatersonar system of sonaphonic transducers or a seismic system of acousticwave transducers or pressure transducers used for oil field exploration.For example, the transducers may be any of the following: antennas, RFsensors, sonaphones, sound sensors, seismic sensors, acoustic wavesensors and/or pressure sensors. Those skilled in the art will recognizeother types of transducer arrays to which the invention is applicable.In general, the invention is applicable to any array having phaseambiguity.

In one embodiment, the invention comprises a system for determining adirection of arrival of a signal (radiation) (radiation) emitted by asource. A first transducer receives the emitted signal (radiation) andprovides a first transducer output signal corresponding to the emittedsignal received by the first transducer. A second transducer is spaced adistance D₁₂ from the first transducer. The second transducer receivesthe emitted signal and provides a second transducer output signalcorresponding to the emitted signal received by the second transducer. Afirst receiver receives the first transducer output signal and providesa first receiver output signal indicating the phase of the firsttransducer output signal received by the first transducer. A secondreceiver receives the second transducer output signal and provides asecond receiver output signal indicating the phase of the secondtransducer output signal received by the second transducer. A processorreceives the first receiver output signal and the second receiver outputsignal, the processor determining a first set of interferometer planescorresponding to a phase difference between the first transducer outputsignal and the second transducer output signal, the phase differencebeing a function of the distance D₁₂. The processor provides outputinformation corresponding to a direction of arrival of the emittedsignal relative to the first and second transducers, wherein the outputinformation is a function of an intersection of the set ofinterferometer planes with a direction cosine sphere.

In another embodiment, the invention comprises a method for determininga direction of arrival of a signal (radiation) emitted by a source, themethod comprising:

receiving the emitted signal with a first transducer and providing afirst transducer output signal corresponding to the emitted signalreceived by the first transducer;

receiving the emitted signal with a second transducer spaced a distanceD₁₂ from the first transducer and providing a second transducer outputsignal corresponding to the emitted signal received by the secondtransducer;

determining a first set of interferometer planes corresponding to aphase difference between the first transducer output signal and thesecond transducer output signal, the phase difference being a functionof the distance D₁₂; and

providing output information corresponding to a direction of arrival ofthe emitted signal relative to the first and second transducers, whereinthe output information is a function of an intersection of the set ofinterferometer planes with a direction cosine sphere.

In another embodiment, a system determines a direction of arrival of asignal (radiation) emitted by a source. Four non-coplanar spacedtransducers receive the emitted signal and provide a transducer outputsignal corresponding to the received, emitted signal. A multi-channelreceiver has each channel associated with one of the transducers toreceive the associated transducer output signal and each channelprovides a digital receiver output signal indicating the phase of thereceived associated transducer output signal. A digital signal processorreceives the digital receiver output signals and processes the receiveddigital receiver output signals by employing a direction findingalgorithm to minimize phase ambiguities between the digital receiveroutput signals to determine a direction of arrival of the emitted signalrelative to the transducers.

In another embodiment, the invention comprises a method for determininga direction of arrival of a signal (radiation) emitted by a source, themethod comprising:

receiving via four non-coplanar, spaced transducers the emitted signaland providing a transducer output signal corresponding to the receivedemitted signal from each transducer;

receiving the associated transducer output signal and providing areceiver output signal indicating the phase of the received emittedsignal; and

processing the received output signals by employing a direction findingalgorithm to minimize phase ambiguities between the received transduceroutput signals to determine a direction of arrival of the emitted signalrelative to the transducers.

In another embodiment, the invention comprises a system for determininga direction of arrival of a signal (radiation) emitted by a source andfor resolving front to back phase ambiguity. A plurality ofnon-collinear, non-coplanar, spaced transducers receives the emittedsignal and provide a transducer output signal corresponding to thereceived emitted signal. A multi-channel receiver has each channelassociated with one of the transducers to receive each associatedtransducer output signal and provides a receiver output signalindicating the phase of the received signal. A processor receives thereceiver output signals and processes the received transducer outputsignals by employing a direction finding algorithm to minimize phaseambiguities between the receiver output signals to determine a first andsecond direction. The processor determines an amplitude comparisonbetween two of the received transducer output signals of the transducerelements and of the received transducer output signal of anothertransducer element receiving from a direction substantially opposite toa direction in which of the two transducers receive. The processorselects the first or the second direction as a function of thedetermined amplitude comparison, the selected direction corresponding tothe direction of arrival of the emitted signal relative to thetransducers.

In another embodiment, the invention comprises a method for determininga direction of arrival of a signal (radiation) emitted by a source andfor resolving front to back phase ambiguity, the method comprising:

receiving the emitted signal via a plurality of non-collinear,non-coplanar, spaced transducers, and providing a transducer outputsignal corresponding to the received emitted signal from eachtransducer;

receiving each associated transducer output signal and providing areceiver output signal indicating the phase of the received emittedsignal;

processing the received transducer output signals by employing adirection finding algorithm to minimize phase ambiguities between thereceived transducer output signals to determine a first and seconddirection;

determining an amplitude comparison between the received transduceroutput signals of two of the transducer elements and the receivedtransducer output signal of another transducer element receiving from adirection substantially opposite to a direction in which of the twotransducers receive;

selecting the first or the second direction as a function of thedetermined amplitude comparison, the selected direction corresponding tothe direction of arrival of the emitted signal relative to thetransducers.

Other objects and features will be in part apparent and in part pointedout hereinafter.

BRIEF DESCRIPTION OF THE DRAWINGS

Having thus described the invention in general terms, reference will nowbe made to the accompanying drawings wherein:

FIG. 1 is an illustration of the geometric relationship of theinterferometer equation (1) between two antennas, A1 and A2, locatedarbitrarily in the Y-Z plane and separated by a distance D₁₂.

FIG. 2A illustrates a Hanson ambiguity diagram for a four elementinterferometer array as illustrated in FIG. 2B.

FIG. 3 is diagram illustrating an ambiguity for the four elementinterferometer array of FIG. 2 and further illustrating the effects ofphase errors on the ambiguity trajectories.

FIG. 4 is an illustration of the relationship between the directioncosine sphere and the interferometer planes showing in dashed lines thetwo circles of intersection of the sphere and the planes and showing thepoint of intersection of the two circles of intersection, according tothe invention.

FIG. 5 is an illustration of a direction cosine sphere with ambiguitycircles; four of the ambiguity circles are intersect at a single point,while the fifth circle is represented as four separate dashed linecircles, each with a different ambiguity integer.

FIG. 6 is an illustration, partially in block diagram form, of a systemof the invention including six surface embedded antenna elements A1-A6on a curved surface.

FIGS. 7A-7D illustrate a procedure according to the invention fordetermining a value of the parameter t that minimizes the value ofD_(ij).

FIGS. 8 and 9 illustrate the process of determining the common point ofintersection.

FIGS. 10 and 11 illustrate the direction vectors from the process ofdetermining the common point of intersection.

Corresponding reference characters indicate corresponding partsthroughout the drawings.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

In general, a collection of a plurality of antennas or transducerswithin a single system is commonly referred to as an array. In thisdescription a practical interferometric system shall comprise of such anarray of antennas or transducers, each element of which is connected toa channel of a multichannel receiver. The connection may be eitherdirectly to a single channel of a multiple channel receiving network orindirectly through a multiplexing switch selection network sharing onechannel of the receiving network amongst two or more antennas.Additionally, such systems are also equipped with a processor forexecuting certain algorithms among which are algorithms specificallydesigned to carry out the phase ambiguity resolution calculations.

It is contemplated that the system and method may be any array oftransducers. For clarity and simplicity, the system and method of theinvention will be described below in the context of an antenna arrayhaving two antenna elements. Those skilled in the art will recognizevarious transducer arrays, including but not limited to light, sound,radio frequency or other radiation transducers, which may be embodied inthe system and/or method of the invention. Those skilled in the art willalso recognize that such arrays may comprise more than two transducers.

The present invention deals with non-coplanar arrangements of antennas,although it will work equally well for co-planar arrangements. FIG. 4 isbased on a pair antennas arbitrarily arranged in a three dimensionalcoordinate system. Unlike the antenna pair in FIG. 1, in thisarrangement the inter-baseline vector has an x-component as well as a y-and a z-component. This x-component alters the interferometer equationfrom that of Eq. 1 so that the new form appears as $\begin{matrix}{{\psi = {{\frac{2\pi\quad d_{x}}{\lambda}\cos\quad\phi\quad\sin\quad\theta} + {\frac{2\pi\quad d_{y}}{\lambda}\sin\quad\phi\quad\sin\quad\theta} + {\frac{2\pi\quad d_{z}}{\lambda}\cos\quad\theta} - {2\pi\quad m}}}{{m = 0},{\pm 1},{\pm 2},{\ldots\quad.}}} & (4)\end{matrix}$

Replacing cos φ sin θ with X, sin φ sin θ with Y, and cos θ with Z, thislast equation can be manipulated into the following form:$\begin{matrix}{{\frac{\lambda}{d}( {\frac{\psi}{2\pi} + m} )} = {{( \frac{d_{x}}{d} )X} + {( \frac{d_{y}}{d} )Y} + {( \frac{d_{z}}{d} ){Z.}}}} & (5)\end{matrix}$

wherein d=√{square root over (d_(x) ²+d_(y) ²+d_(z) ²)}. This is theequation of a set of parallel planes, one plane for each value of theambiguity integer m; these planes as well as whole sets of these planesare referred to as interferometer planes. The normal to the surfaces ofthese interferometer planes is given by $\begin{matrix}{n = {{\frac{d_{x}}{d}i} + {\frac{d_{y}}{d}j} + {\frac{d_{z}}{d}{k.}}}} & (6)\end{matrix}$

The unit vector extending from the origin of configuration spacecoordinate system in the direction of the distant emitter is given bythe expression{circumflex over (r)} cos φ sin θ{circumflex over (i)}+sin φ sinθ{circumflex over (j)}+cos θ{circumflex over (k)}.  (7)

Equation (7) in light of the substitutions above can be recast into thedirection cosine space expression:{circumflex over (r)}=Xî+Yĵ+Z{circumflex over (k)}  (8)

This unit vector also defines a sphere in direction cosine space, theradius of which is exactly 1:X ² +Y ² +Z ²=1.  (9)

This sphere is referred to as the direction cosine sphere and the planesdefined by Eq. 5 will intersect this sphere in a set of curves and thedistant emitter must lay along a line in direction cosine space thatbegins at the origin and extends outward through the surface of thesphere along one of these curves of intersection. As shown in FIG. 4these curves of intersection of a plane with a sphere are always eithera circle or a mere point, a point being a degenerate circle the radiusof which is exactly zero. This circular curve of intersection isreferred to as an ambiguity circle, e.g., circle #1 and circle #2.

The distance, d_(n), between the origin of the direction cosine spacecoordinate system and the plane defined by Eq. 5 along the normal vectordefined in Eq. 6 is given by $\begin{matrix}{d_{n} = {\frac{\lambda}{d}{( {\frac{\psi}{2\pi} + m} ).}}} & (10)\end{matrix}$

The allowed values of the ambiguity integer m are those for which d_(n)falls within −1 and +1. For if m were chosen so as to make d_(n) greaterthan 1 (or less than −1), the corresponding interferometer plane wouldnot then intersect with the direction cosine sphere. While suchinterferometer planes exist mathematically, they are said to reside ininvisible space. It is apparent from this last equation that theambiguity integer m is bounded within the range $\begin{matrix}{m = {\pm {{{INT}\lbrack {\frac{d}{\lambda} + \frac{1}{2}} \rbrack}.}}} & (11)\end{matrix}$

Reference is made again to FIG. 4 where two interferometer planes areillustrated intersecting with the direction cosine sphere each forming aseparate ambiguity circle, e.g., circle #1 and circle #2. These twocircles are created by an interferometer consisting of three antennas.Although not shown in FIG. 4, each pair of antennas, if theinter-element distance is great enough, will give rise to several suchambiguity circles each, and some of the ambiguity circles of one pairwill undoubtedly intersect with some of the ambiguity circles of theother pair, creating multiple points of intersection or angularambiguities. A practical interferometer will include four or moreantennas that can be arranged so that only a single ambiguity circlefrom each set will intersect at a single common point.

Reference is made to FIG. 5 where the interferometer system consists ofsix antennas creating five sets of ambiguity circles. In thisillustration a common point of intersection has been found for the firstfour sets of ambiguity circles and the phase ambiguity resolutionalgorithm is testing the four circles of the fifth set to see whichpasses through (or closest to) the previously determined common point ofintersection for the first four sets.

When the antennas are arrayed in a non-coplanar manner, there is but onecommon point of intersection for the proper ambiguity circles of all ofthe sets. This point is in effect “in front” of the interferometer arrayantennas in that this point and the emitter both reside in the sameforward hemisphere where X is positive. When the antennas are arrangedas a coplanar array, there are two points of intersection of the properambiguity circles of all of the sets, one point in the forwardhemisphere and yet a second in the rear hemisphere. The prior artcontains several method for resolving this “front-back” ambiguitycustomarily involving a preferentially rearward facing antenna. In thisway, the signal (radiation) amplitude received by this rearward facingantenna is compared with the signal (radiation) amplitude received byone of the “forward facing” interferometer antennas. If the signalamplitude received by the rearward facing antenna is greater than thesignal amplitude received by the interferometer antenna, the emitter is“declared to be behind the interferometer and all direction findingprocessing halted for that signal. In the remainder of this applicationthe discussion will proceed as if there is but a single point ofintersection for two ambiguity circles, when in fact there are alwaystwo such points—one in the forward hemisphere and one in the rearhemisphere. It is to be understood that the algorithms being discussedhave the primary objective of finding only the point of intersection inthe forward hemisphere.

FIG. 6 illustrates the preferred embodiment of an interferometer system600 according to the invention including a non-coplanar antenna array602 on an aircraft skin 604. In this FIG. 6, the interferometer array602 is shown with a total of six antennas A1-A6, all laying on a commoncomplex surface which is assumed for exemplary purposes only to be anaircraft skin 604. As shown in FIG. 6, the RF signal paths out of theseantennas go through an RF distribution network 606 where the signalsfrom the antennas A1-A6 are shared amongst the channels of a phasesensitive, multi-channel electronic support measures receiver 608.Optionally, the number of receiver channels may and (and is assumed tobe) fewer than the number of antennas. Following the extraction ofincident signal's characteristics (RF frequency, pulse width, signalamplitude and channel-channel phase differences) by the electronicsupport measures receiver 608, digital representations of thesecharacteristics are past to a digital signal processor 610 where, amongother processes, ambiguity resolution and angle of arrival processingtake place.

In one embodiment, the digital signal processor 610 receives digitalreceiver output signals indicating the phase of the signals received bythe antennas and processes the received digital receiver output signalsby employing a direction finding algorithm to minimize phase ambiguitiesbetween the digital receiver output signals to determine a direction ofarrival of the emitted signal relative to the antennas.

As a result, system 600 determines a direction of arrival of a signal(radiation) emitted by a source which is remote from array 602. Thearray 600 includes at least a first antenna A1 for receiving the emittedsignal and for providing a first antenna output signal 612 correspondingto the emitted signal received by the first antenna. The array alsoincludes at least a second antenna A2 spaced a distance D₁₂ from thefirst antenna A1. The second antenna A2 also receives the emitted signaland provides a second antenna output signal 614 corresponding to theemitted signal received by the second antenna. The multi-channelreceiver 608 includes a first receiver for receiving the first antennaoutput signal 612 and for providing a first receiver output signal(e.g., digital signal data) indicating the phase of the first antennaoutput signal received by the first antenna (and possibly indicating thecarrier). The multi-channel receiver 608 includes a second receiver forreceiving the second antenna output signal 614 and for providing asecond receiver output signal (e.g., digital signal data) indicating thephase of the second antenna output signal received by the second antenna(and possibly indicating the carrier). The processor 610 receives thefirst receiver output signal and the second receiver output signal anddetermines a first set of interferometer planes corresponding to a phasedifference between the first antenna output signal and the secondantenna output signal. As explained herein, the phase difference is afunction of the distance D₁₂. The processor 610 provides outputinformation corresponding to a direction of arrival of the emittedsignal relative to the first and second antennas. As explained herein,the output information is a function of an intersection of the set ofinterferometer planes with a direction cosine sphere.

In one embodiment, the first receiver output signal has a phasecorresponding to the phase of the signal received by the first antennaand the second receiver output signal has a phase corresponding to thephase of the signal received by the second antenna.

In one embodiment as described herein, the system resolves front to backphase ambiguity. The processor 610 processes the received antenna outputsignals by employing a direction finding algorithm to minimize phaseambiguities between the receiver output signals to determine a first andsecond direction. The processor determines an amplitude comparisonbetween two of the received antenna output signals of the antennaelements and of the received antenna output signal of another antennaelement (not shown) receiving from a direction substantially opposite toa direction in which of the two antennas receive. The processor selectsthe first or the second direction as a function of the determinedamplitude comparison, the selected direction corresponding to thedirection of arrival of the emitted signal relative to the first andsecond antennas.

Optionally, the system 600 may include additional antennas such as athird antenna A3 receiving the emitted signal and providing a thirdantenna output signal corresponding to the emitted signal received bythe third antenna A3. The third antenna is spaced a distance D₂₃ fromthe second antenna A2. The receiver 608 includes a third receiver forreceiving the third antenna output signal and providing a third receiveroutput signal indicating the phase of the third antenna output signalreceived by the third antenna A3. In this optional embodiment, theprocessor receives the third receiver output signal and determines asecond set of interferometer planes corresponding to a phase differencebetween the second antenna output signal and the third antenna outputsignal. As described herein, the phase difference is a function of thedistance D₂₃. The processor provides output information corresponding toa direction of arrival of the emitted signal relative to the first,second and third antennas A1-A3. As described herein, the outputinformation is a function of an intersection of the second set ofinterferometer planes with a second direction cosine sphere.

The invention also includes the method for determining the direction ofarrival of the signal (radiation) emitted by a source. The methodcomprises:

receiving the emitted signal with the first antenna A1;

receiving the emitted signal with the second antenna A2; determining afirst set of interferometer planes corresponding to a phase differencebetween the emitted signals; and

providing output information corresponding to a direction of arrival ofthe emitted signal relative to the first and second antennas, whereinthe output information is a function of an intersection of the set ofinterferometer planes with a direction cosine sphere.

One embodiment includes a method for determining a direction of arrivalof a signal (radiation) emitted by a source and for resolving front toback phase ambiguity, wherein the method comprises:

processing the received antenna output signals by employing a directionfinding algorithm to minimize phase ambiguities between the receivedantenna output signals to determine a first and second direction;

determining an amplitude comparison between the received antenna outputsignals of two of the antenna elements and the received antenna outputsignal of another antenna element receiving from a directionsubstantially opposite to a direction in which of the two antennasreceive; and

selecting the first or the second direction as a function of thedetermined amplitude comparison, wherein the selected directioncorresponds to the direction of arrival of the emitted signal relativeto the antennas.

Another embodiment includes an iterative method of arranging a pluralityof antennas (particularly non-collinear, non-coplanar spaced antennas)having phase errors in order to use the antennas to determine adirection of arrival of a signal (radiation) emitted by a source. Inthis iterative method, a set of circle pair intersections for each pairof the plurality of antennas is determined. The relative position of theantennas is then modified (e.g., by modifying the spacing) and amodified set of circle pair intersections for each pair of the pluralityof antennas after modifying the position is determined. The set ofcircle pair intersections is compared with the modified set of circlepair intersections to determine the more tightly grouped set of circlepair intersections. The antennas are arranged according the positionwhich corresponds to the more tightly grouped set of circle pairintersections. The resulting antenna array also embodies the invention.The interactive method may instead include and may also includemodifying the preset number of the antennas (e.g., adding or removingantennas from the array) and determining a modified set of circle pairintersections for each pair of the plurality of antennas after modifyingthe preset number. The set of circle pair intersections before modifyingis compared with the modified set of circle pair intersections todetermine the more tightly grouped set of circle pair intersections. Thearray is then assembled with a number of antennas which number whichcorresponds to the more tightly grouped set of circle pairintersections. The resulting antenna array also embodies the invention.

The “Best” Set of Pair-Wise Intersections

In the present case, the objective of the ambiguity resolution algorithmis to determine the common point of intersection of the variousambiguity circles. An actual algorithm will involve a set of nestedloops each of which iterates through the allowed values of the ambiguityinteger associated with each pair of array antennas. In this manner,every combination of the ambiguity circles of all the several sets areexamined until the “best” set is discovered by comparison.

In practical interferometers thermal noise disturbances and othersources of phase error quite often prevent the proper set of ambiguitycircles from all crossing at a single common point. In these cases then,some criteria is established, such as the most tightly grouped set ofpair-wise points of intersection, which when found is declared to be the“best” set of pair-wise intersections from which a common point can bederived. The following is the description of the preferred algorithmdesigned to find the most tightly grouped set of ambiguity circleintersection points. It is pointed out here that the single mostimportant aspect of this algorithm is the determination of the point ofintersection of two ambiguity circles from separate sets ofinterferometer planes. The following description explains someembodiments as to how this objective is accomplished.

The first step is to determine the radius of the j^(th) ambiguity circleof the i^(th) set of interferometer planes. (The algorithm describedherein below is illustrated by the flow charts contained in FIGS. 11through 14.) By the Pythagorean theorem if the normal distance from thesurface of the interferometer plane back to the origin is given byEquation 10, then the radius of the j^(th) ambiguity circle is given by$\begin{matrix}{R_{ij} = {\sqrt{1 - d_{ni}} = {\sqrt{1 - \{ {\frac{\lambda}{d_{ni}}( {\frac{\psi_{i}}{2\pi} + m_{ij}} )} \}^{2}}.}}} & (12)\end{matrix}$

The next step is to recognize that the normal vector to the i^(th)interferometer plane provides the basis for defining a coordinate systemin which the ambiguity circles of the i^(th) interferometer planes aremost simply defined. This normal vector is given in Eq. 6 and isrepeated here for completeness: $\begin{matrix}{n = {{\frac{d_{x}}{d}i} + {\frac{d_{y}}{d}j} + {\frac{d_{z}}{d}{k.}}}} & (6)\end{matrix}$

Let this vector be the unit vector in the direction of the z-axis inthis new coordinate system—referred to hereafter as the primedcoordinate system. Form the unit vector in the direction of y′ bysetting the x-axis component equal to zero and interchanging the y andthe z components and negating the new y′ component: $\begin{matrix}{{z^{\prime} = {{\frac{d_{x}}{d}i} + {\frac{d_{y}}{d}j} + {\frac{d_{z}}{d}k}}},{and}} & ( {13a} ) \\{{y^{\prime} = {{{- \frac{d_{z}}{d^{\prime}}}j} + {\frac{d_{y}}{d^{\prime}}k}}}{wherein}{d^{\prime} = {\sqrt{d_{y} + d_{z}}.}}} & ( {13b} )\end{matrix}$

The vector cross product of these two vectors gives the unit vectoraligned with the x′-axis. $\begin{matrix}{x^{\prime} = {{y^{\prime} \times z^{\prime}} = {\frac{1}{{dd}^{\prime}}{\{ {{{- ( {d_{z}^{2} + d_{y}^{2}} )}\hat{i}} + {( {d_{x}d_{y}} )\hat{j}} + {( {d_{x}d_{z}} )\hat{k}}} \}.}}}} & ( {13c} )\end{matrix}$

These three unit vectors form a basis set for a Cartesian coordinatesystem that is at once lined up with the normal vector to theinterferometer plane with its x′-y′ coordinate axes residing in theinterferometer plane. Vector transformations from the primed to theunprimed coordinate systems can be accomplished using the 3-by-3 matrix$\begin{matrix}{T = {\frac{1}{{dd}^{\prime}}{\begin{pmatrix}{- ( {d_{y}^{2} + d_{z}^{2}} )} & 0 & {d^{\prime}d_{x}} \\{d_{x}d_{y}} & {- {dd}_{z}} & {d^{\prime} \cdot d_{y}} \\{d_{y}d_{z}} & {dd}_{y} & {d^{\prime}d_{z}}\end{pmatrix}.}}} & (14)\end{matrix}$

Now, in this new primed coordinate system the parametric equation of thecircle of intersection of i^(th) interferometer plane with the directioncosine sphere is easily seen to be $\begin{matrix}{{r_{i}(t)} = {{\lbrack {R_{i}{\cos(t)}} \rbrack{\hat{i}}^{\prime}} + {\lbrack {R_{i}{\sin(t)}} \rbrack{\hat{j}}^{\prime}} + {\frac{\lambda}{d_{ni}}( {\frac{\Psi_{i}}{2\pi} + m_{ij}} ){{\hat{k}}^{\prime}.}}}} & (15)\end{matrix}$

This parametric form of the ambiguity circle is transformed to theunprimed coordinate system using the transformation matrix of Eq. 13.

The ambiguity circles of a second or subsequent interferometer planeresults in a similar equation within a double primed coordinate system:$\begin{matrix}{{r_{k}(s)} = {{\lbrack {R_{k}{\cos(s)}} \rbrack{\hat{i}}^{''}} + {\lbrack {R_{k}{\sin(s)}} \rbrack{\hat{j}}^{''}} + {\frac{\lambda}{d_{nk}}( {\frac{\Psi_{n}}{2\pi} + m_{kj}} ){{\hat{k}}^{''}.}}}} & (16)\end{matrix}$

and again a similar transformation matrix as in equation 13 transformsthis vector equation from the doubly primed coordinate system to theunprimed coordinate system, allowing the two parametric equations can becalculated in the same coordinate system.

To find the common point between the two ambiguity circles, theparameter s is held constant at some convenient value while theparameter t is varied in small increments over a range of values thatkeep the x-component of r_(i) positive, thus assuring that the commonpoint of intersection, were it to be found, would be in the forwardhemisphere. The search procedure is illustrated in FIGS. 7A-7D.Initially, the process as shown in FIG. 7B looks for a value of theparameter t that minimizes the value of D_(ij). $\begin{matrix}{D_{ij} = {\sqrt{\{ {{x( t_{i} )} - {x( s_{j} )}} \}^{2} + \{ {{y( t_{i} )} - {y( s_{j} )}} \}^{2} + \{ {{z( t_{i} )} - {z( s_{j} )}} \}^{2}}.}} & (17)\end{matrix}$

At this point reference is made to FIG. 8, where as illustrated t_(i+3)is the point that minimizes D_(ij). The search over t is halted at thispoint and the value of the parameter t retained at this value.

Next, the process of the algorithm as shown in FIG. 7C conducts a searchover the parameter s (varied in small increments as was the parametert), again looking for the value of s that minimizes the value of D_(ij)(see FIG. 9). The procedure then reverts back to varying the parameter twhile holding s at its previously determined value. In this process, theincrements of t are taken to be somewhat smaller than the incrementsused in the first process. The procedure continues in this repeatingfashion, making the increments successively smaller and smaller, untilthe value of D_(ij) is less than a pre-determined small value denoted bythe parameter ε in Procedure #3 illustrated in FIG. 7D. The valueassigned to ε is chosen to establish the accuracy with which the commonpoint of intersection is determined in this multi-step procedure. On theother hand, if when D_(ij) is found not to change by an appreciableamount from one step to the next and its value is not very nearly zero,the two ambiguity circles in question are understood not to intersect.

Once common points of intersection for each pair of ambiguity circleshave been found, vectors are formed for each point, each vectororiginating at the origin of the coordinate system and passing throughthe corresponding point of intersection of the two ambiguity circles.For an interferometer system consisting of n antennas, there are n−1unique pairs giving rise to n−1 sets of ambiguity circles. The n−1 setsambiguity circles give rise to no more than a total of ½ (n−1)(n−2)points of intersection and the same number of vectors; some circles ofone set may not be well positioned to intersect with a particular circleof another set. For example six antennas give rise to five unique phasedifferences, and thus five sets of interferometer planes or five sets ofambiguity circles: thus no more than ten points of intersection.

A second method of finding the common point of intersection of twoambiguity circles relies upon projecting the two ambiguity circles ontothe plane that contains the normal vectors of both interferometerplanes. In this plane the two ambiguity circles are each reduced tostraight lines and their common point of intersection is easily foundusing techniques familiar from high school algebra. The normal to thisplane is found as the vector cross product of the normal vectors of theinterferometer planes associated with the two ambiguity circles. Theexact process is similar to the process described above: assign one ofthe two normal vectors to be the z′-axis unit vector for this newcoordinate system; since this normal vector is already a unit vector nofurther normalization is required. Then assign the vector developed fromthe cross product of the two interferometer plane normal vectors to bethe y′-axis unit vector of this new coordinate system once it has beenproperly normalized. Finally assign the cross product of this y′-axisunit vector with the z′-axis unit vector, specifically ŷ′×{circumflexover (z)}′, to be the x′-axis unit vector once this vector too has beenproperly normalized.

In the next step, a coordinate system aligned with the second normalvector is developed exactly as described above using Equation 6 throughEquation 13; this coordinate system will now be the double primedcoordinate system. The parametric vector equation for the ambiguitycircle associated with this second coordinate system is transformed tothe primed coordinate system by first transforming it from the doubleprimed coordinate system to the unprimed coordinate system and then, inturn, transforming it from the unprimed coordinate system to the primedcoordinate system all in one step by a double matrix multiplication: towithR′ ₂(s)=T ₁ ^(T) ·T ₂ ·R″ ₂(s).  (18)

The projection of R′₂(s) onto the y′-z′ plane is accomplished by formingthe vector inner product of R′₂(s) with a unit vector constructed fromthe unit vectors that parallel the y′ and the z′ axes. The point wherethis projection crosses the y′ axis is the point where the two ambiguitycircles intersect but in this primed coordinate system. Transformingthis point back to the unprimed coordinate system produces thecoordinates of the common point of intersection on the surface of thedirection cosine sphere.

Once the common points of intersection for all possible combinations ofthe set of ambiguity circles have been found, it is possible to formdirection vectors that extend from the unprimed coordinate system originout through each point of intersection on the surface of the directioncosine sphere. It is noted that if all of these vectors are tightlygrouped, the sum of the inner products is sure to be close to(⅛)n²(n−1)²−(¼)n(n−1) where n is the number of direction vectors. On theother hand if one or more of these points are not contained within thisgroup, the sum of the inner products is sure to be less than thisnumber. The former case is illustrated in FIG. 10 and the latter in FIG.11.

In one form, as noted above, the invention comprises a system fordetermining a direction of arrival of an rf signal emitted by a rfsource. A first transducer (e.g., antenna A1) receives the emitted rfsignal and provides a first transducer output signal corresponding tothe emitted rf signal received by the first transducer. A secondtransducer (e.g., antenna A2) spaced a distance D₁₂ from the firsttransducer receives the emitted signal and provides a second transduceroutput signal corresponding to the emitted signal received by the secondtransducer. A first receiver (e.g., a channel of receiver 608) receivesthe first transducer output signal and provides a first receiver outputsignal indicating the phase of the first transducer output signalreceived by the first transducer. A second receiver (e.g., anotherchannel of receiver 608) receives the second transducer output signaland provides a second receiver output signal indicating the phase of thesecond transducer output signal received by the second transducer. Aprocessor (e.g., DSP 610) receives the first receiver output signal andthe second receiver output signal and determines a first set ofinterferometer planes corresponding to a phase difference between thefirst transducer output signal and the second transducer output signal.The phase difference is a function of the distance D₁₂, and processorprovides output information corresponding to a direction of arrival ofthe emitted signal relative to the first and second transducers. Theoutput information is a function of an intersection of the set ofinterferometer planes with a direction cosine sphere.

When introducing elements of the present invention or the preferredembodiment(s) thereof, the articles “a”, “an”, “the” and “said” areintended to mean that there are one or more of the elements. The terms“comprising”, “including” and “having” are intended to be inclusive andmean that there may be additional elements other than the listedelements.

In view of the above, it will be seen that the several objects of theinvention are achieved and other advantageous results attained.

As various changes could be made in the above constructions, products,and methods without departing from the scope of the invention, it isintended that all matter contained in the above description and shown inthe accompanying drawings shall be interpreted as illustrative and notin a limiting sense.

1. A system for determining a direction of arrival of a signal emittedby a source and for resolving front to back phase ambiguity, said systemcomprising: a plurality of non-collinear, non-coplanar, spacedtransducers, each receiving the emitted signal and providing atransducer output signal corresponding to the received emitted signal; amulti-channel receiver, each channel associated with one of thetransducers for receiving each associated transducer output signal andproviding a receiver output signal indicating the phase of the receivedsignal; a processor for receiving the receiver output signals and forprocessing the received transducer output signals by employing adirection finding algorithm to minimize phase ambiguities between thereceiver output signals to determine a first direction and a seconddirection, said processor for determining an amplitude comparisonbetween two of the received transducer output signals of the transducerelements and of the received transducer output signal of anothertransducer element receiving from a direction substantially opposite toa direction in which of the two transducers receive, and said processorselecting the first direction or the second direction as a function ofthe determined amplitude comparison, said selected directioncorresponding to the direction of arrival of the emitted signal relativeto the transducers.
 2. The system of claim 1 wherein the transducers areantennas mounted on a surface of an aircraft.
 3. The system of claim 1wherein the transducers are selected from the following: antennas, rfsensors, sonaphones, sound sensors, seismic sensors, acoustic wavesensors and/or pressure sensors.
 4. The system of claim 1 wherein: afirst channel of the receiver receives a first transducer output signaland provides a first channel output signal having a phase correspondingto the phase of the signal received by the first transducer; a secondchannel of the receiver receives the second transducer output signal andprovides a second channel output signal having a phase corresponding tothe phase of the signal received by the second transducer; and theprocessor receives the first channel output signal and the secondchannel output signal and provides an output signal corresponding to thedirection of arrival.
 5. The system of claim 4 wherein the plurality oftransducers comprises a first transducer and a second transducer andfurther comprising: a third transducer receiving the emitted signal andproviding a third transducer output signal corresponding to the emittedsignal received by the third transducer, said third transducer spaced adistance D₂₃ from the second transducer; a third channel of the receiverfor receiving the third transducer output signal and providing a thirdchannel output signal indicating the phase of the third transduceroutput signal received by the third transducer; and said processorreceives the third channel output signal, said processor determining aset of interferometer planes corresponding to a phase difference betweenthe second transducer output signal and the third transducer outputsignal, said phase difference being a function of the distance D₂₃, andsaid processor providing output information corresponding to a directionof arrival of the emitted signal relative to the first, second and thirdtransducers, wherein the output information is a function of anintersection of the set of interferometer planes with a direction cosinesphere.
 6. The system of claim 5 wherein said processor determines saidamplitude comparison as a function of a distance D₂₃ between the secondtransducer and the third transducer.
 7. The system of claim 6 whereinthe transducers are antennas mounted on a surface of an aircraft.
 8. Thesystem of claim 1 wherein said processor determines said amplitudecomparison as a function of a distance between two of said plurality oftransducers.
 9. A method for determining a direction of arrival of asignal emitted by a source and for resolving front to back phaseambiguity, said method comprising: receiving the emitted signal via aplurality of non-collinear, non-coplanar, spaced transducers, andproviding a transducer output signal corresponding to the receivedemitted signal from each transducer; receiving each associatedtransducer output signal and providing a receiver output signalindicating the phase of the received emitted signal; processing thereceived transducer output signals by employing a direction findingalgorithm to minimize phase ambiguities between the received transduceroutput signals to determine a first direction and a second direction;determining an amplitude comparison between the received transduceroutput signals of two of the transducer elements and the receivedtransducer output signal of another transducer element receiving from adirection substantially opposite to a direction in which of the twotransducers receive; selecting the first direction or the seconddirection as a function of the determined amplitude comparison, saidselected direction corresponding to the direction of arrival of theemitted signal relative to the transducers.
 10. The method of claim 9wherein the transducers are antennas mounted on a surface of anaircraft.
 11. The method of claim 9 wherein the transducers are selectedfrom the following: antennas, rf sensors, sonaphones, sound sensors,seismic sensors, acoustic wave sensors and/or pressure sensors.
 12. Themethod of claim 9 wherein: receiving a first transducer output signaland providing a first output signal having a phase corresponding to thephase of the received first transducer output signal; receiving a secondtransducer output signal and providing a second output signal having aphase corresponding to the phase of the second transducer output signal;and receiving the first output signal and the second output signal andproviding an output signal corresponding to the direction of arrival.13. The method of claim 12 wherein the plurality of transducerscomprises a first transducer and a second transducer, and furthercomprising: receiving the emitted signal with a third transducer andproviding a third transducer output signal corresponding to the emittedsignal received by the third transducer; determining a second set ofinterferometer planes corresponding to a phase difference between thesecond transducer output signal and the third transducer output signal,providing output information corresponding to a direction of arrival ofthe emitted signal relative to the first, second and third transducers,wherein the output information is a function of an intersection of thesecond set of interferometer planes with a second direction cosinesphere.
 14. The method of claim 13 wherein said determining determinessaid amplitude comparison as a function of a distance D₂₃ between thesecond transducer and the third transducer.
 15. The method of claim 14wherein the transducers are antennas mounted on a surface of anaircraft.
 16. The method of claim 9 wherein said determining determinessaid amplitude comparison as a function of a distance between two ofsaid plurality of transducers.